Lemma 2.3 (Lemma 5.1 of [1]})
Let \(\tilde{\sigma }= \sqrt{ \tilde{\lambda }}\)  be a singular value of \(\frac{1}{\sqrt{m}} \widetilde{X}_n\)  that is not a singular value of \(\frac{1}{\sqrt{m}} X_n\)  and let \(\tilde{u}, \tilde{v}\)  be a corresponding pair of singular unit vectors. Then the column vector
\( \hspace{-199.16928pt} \begin{bmatrix} \Theta V_m^{\prime } \tilde{v} \\ \Theta U_n^{\prime } \tilde{u} \end{bmatrix} \hspace{199.16928pt} \nonumber \) 
